How to find the Area of Trapezoids? – Formula, Examples, Derivation

What is a trapezoid? A trapezoid or trapezium is a quadrilateral with at least one pair of parallel sides. The parallel sides are known as bases. When the other two sides are non-parallel, they are called legs or lateral sides. Else there are two pairs of bases. Real-life examples where you can see the area of trapezoids are handbags, popcorn tins, and the guitar-like dulcimer. The area of a trapezoid is the complete space enclosed by its four sides. There are two approaches to finding the area of trapezoids. 

1. The first method is a direct method that uses a direct formula to find the area of a trapezoid with the known dimensions (see example 1)
2. For the second method, firstly, if we are given the length of all the sides, we split the trapezoid into smaller polygons such as triangles and rectangles. Next, we will find the area of the triangles and rectangles separately. Finally, we will add the area of the polygons to get the total area of the trapezoid. (see example 2 for  a more precise understanding)

What is the Formula To Calculate the Area of Trapezoids?

We can calculate the area of a trapezoid if we know the length of its parallel sides and the distance (height) between the parallel sides. The area of the trapezoids formula is:   

A = ½ (a + b) h

Here (A) is the area of a trapezoid.

‘a’ and ‘b’ are the parallel sides of the trapezoid

‘h’ is the height, i.e., the perpendicular distance between the parallel sides.

Area of Trapezoid Example 

Here is an area of a trapezoid example using the direct formula and an area of a trapezoid example with the alternative method.

Example 1: Find the area of a trapezoid given the length of parallel sides 22 cm and 12 cm, respectively. The height is 5 cm. Solution: Given: The bases are : a = 22 cm; b = 10 cm; the height is h = 5 cm. The area of the trapezoid = A = ½ (a + b) h A = ½ (22 + 10) × (5)  A = ½ (32) × (5)  A = ½  × 160 A = 80 cm2

Example 2: Find the area of a trapezoid whose parallel sides are given as 10cm and 16cm, respectively, and the non-parallel sides are 5cm each. Solution: Since in this question, we don’t have the height of the trapezium, we will follow the following steps to calculate the area of the trapezoid. Given: a =10 cm; b =16 cm; non-parallel sides = 5 cm each Step 1: To find the height of the trapezoid, we will first draw the height of the trapezoid on both sides. Now we can see that the trapezoid consists of a rectangle ABQP and 2 right-angled triangles, APD and BQC. Step 2: Now, we have to find the length of DP and QC.  Since ABQP is a rectangle, AB = PQ DC = 16 cm (Given) So, PQ = AB We can find the combined length of DP + QC as follows DC – PQ = 16 – 10 = 6 cmSo, DP + QC = 6 6 ÷ 2 = DP = QC  3 cm = DP = QC Step 3: AP = BQ (opposite and equal sides of a rectangle)   AD = BC = 5 cm (Given) So, we can calculate the height AP and BQ using Pythagoras theorem. In the right-angled triangle ADP  AP = √(AD2 – DP2)  AP = √(52 – 32)  AP = √(25 – 9) = √16 = 4 cm Since ABQP is a rectangle, the opposite sides will be equal. So, AP = BQ = 4 cm.  Step 4: Now we know all the dimensions of the trapezoid. We can calculate the area using the formula. Area of a trapezoid = ½ (a + b) h; where h = 4 cm, a =10 cm b = 16 cm On substituting values we get: Area = ½ (10 + 16) × 4 Area = ½ × 26 × 4 Area = 52 cm2 We can calculate by adding area of the rectangle and two triangles Area of trapezoid = Area of ABPQ + Area of ADP + Area of BQC Area of trapezoid = (l × b) + 2( ab/2) Area of trapezoid = (10 × 4) + 2(3 ×4/2) A = 40 + 12 A = 52 cm2

Trapezoidal Prism

A trapezoidal prism is a 3D figure made up of two congruent trapezoids that are connected by four rectangles. The trapezoids are on the top and the bottom. Thus, they form the base for prisms and have polygons that form their bases. The four rectangles form the lateral faces of the trapezoid prism. So, a trapezoidal prism consists of-

• Six faces
• Eight vertices
• Twelve edges

Surface Area of a Trapezoidal Prism

The area of a trapezoidal prism is the sum of the surfaces of the prism. This area equals the areas of all of the faces of the trapezoid prism. Since a trapezoidal prism has two trapezoidal faces and four rectangular faces, a sum of their areas will give the surface area of the prism. However, there is a simple and direct formula for calculating the surface area of a trapezoid prism. The formula is:

Surface Area of a Trapezoidal Prism = h (b + d) + l (a + b + c + d) square units. 

Here, h = height

b and d are the lengths of the base

a + b + c + d is the perimeter

l is the lateral surface area of a trapezoidal prism

Derivation of Surface Area of a Trapezoidal Prism 

 

The base of a trapezoidal prism is trapezoid in shape. Here,

b and d are parallel sides of the trapezoid

H = distance between the parallel sides 

l = length of the trapezoidal prism

So, the total surface area of a trapezoidal prism (TSA) = 2 × areas of base + lateral surface area ————- (1)

Area of a trapezoid =  ½ (base 1 + base 2) height

So, area of trapezoidal base = h (b + d)/2 ——————— (2)

The lateral area of a trapezoidal prism (LSA) = sum of the areas of each rectangular surface

So, LSA = (a × l) + (b × l) + (c × l) + (d × l) ——————- (3)

On substituting the values from equation (2) and equation (3) in equation 1 i.e., TSA formula, we get: 

(TSA) = 2 × h (b + d)/2 + (a × l) + (b × l) + (c × l) + (d × l) 

TSA = h (b + d) + [(a × l) + (b × l) + (c × l) + (d × l)]

Total surface area of a trapezoidal prism = h (b + d) + l (a + b + c + d)

Thus, TSA of a trapezoidal prism = h (b + d) + l ( a + b + c + d) unit square

How to Find the Surface Area of a Trapezoidal Prism?

Here are the steps on how to find the surface area of a trapezoidal prism.

Step 1: Spot the four sides of the trapezium – a, b, c, and d. These represent the widths of the four rectangles. The addition of these 4 values will give the perimeter P.

Step 2: Spot the length h of the prism.

Step 3: Find the lateral area of a trapezoidal prism.

Step 4: Identify b1, b2 and h of the trapezoid. Now, find the base area B using the formula, (b1 + b2) h/2

Step 5: Lastly, put the values in the formula = 2B + Lateral Surface Area to calculate the total surface area of the trapezoidal prism.

Consider the following example of the surface area of a trapezoidal prism for a clearer understanding.

Example 1:  Find the total surface area of an isosceles trapezoidal prism that has parallel edges of the base 8 cm and 12 cm. The legs of the base are 5 cm each, the altitude of the base is 4 cm and the height of the prism is 10 cm. Solution: The perimeter of the base = sum of the length of the sides. p=8+5+12+5=30  cm Given that the base is an isosceles trapezoid, therefore, the area = 1/2h(b1+b2)  Area of base =1/2(4)(8+12)=40 cm2 T.S.A.= ph+2B   T.S.A = 30(10)+2(40) T.S.A =300+80 =380  cm2